Past Functional Analysis Seminar

To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. 

Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin. 

The (finite) Hilbert matrix is arguably one of the single most well-known matrices in mathematics. The infinite Hilbert matrix H was introduced by David Hilbert around 120 years ago in connection with his double series theorem. It can be interpreted as a linear operator on spaces of analytic functions by its action on their Taylor coefficients. The boundedness of H on the Hardy spaces H^p for 1 < p < ∞ and Bergman spaces A^p for 2 < p < ∞ was established by Diamantopoulos and Siskakis. The exact value of the operator norm of H acting on the Bergman spaces A^p for 4 ≤ p < ∞ was shown to be π /sin(2π/p) by Dostanic, Jevtic and Vukotic in 2008. The case 2 < p < 4 was an open problem until in 2018 it was shown by Bozin and Karapetrovic that the norm has the same value also on the scale2 < p < 4. In this talk, we introduce some background, review some of the old results, and consider the still partly open problem regarding the value of the norm on weighted Bergman spaces. We also consider a generalised Hilbert matrix operator and its (essential) norm. The talk is partly based on a joint work with Mikael Lindström, David Norrbo, and Niklas Wikman (Åbo Akademi University).
 

I will present novel freeness results in ultraproducts of tracial von Neumann algebras. As a particular case, I will show that if a and b are the generators of the free group F_2, then the relative commutants of a and b in the ultraproduct of the free group factor are free with respect to the ultraproduct trace. The proof is based on a surprising application of Lp-boundedness results of Fourier multipliers in free group factors for p > 2. I will describe applications of these results to absorption and model theory of II_1 factors. This is joint work with Adrian Ioana.

In ring theory, Morita equivalence is an invariant for many properties, generalising the isomorphism of commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years, there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphisms, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.

This talk concerns some ideas around the question of when a *-homomorphism into a quotient C*-algebra lifts. Lifting of *-homomorphisms arises prominently in the notions of projectivity and semiprojectivity, which in turn are closely related to stability of relations. Blackadar recently defined the notions of l-open and l-closed C*-algebras, making use of the topological space of *-homomorphisms from a C*-algebra A to another C*-algebra B, with the point-norm topology. I will discuss these properties and present new characterizations of them, which lead to solutions of some problems posed by Blackadar. This is joint work with Dolapo Oyetunbi.

A homogeneous space G/H is called a reductive symmetric space if G is a (real) reductive Lie group, and H is a symmetric subgroup of G, meaning that H is the subgroup fixed by some involution on G. The representation theory on reductive symmetric spaces was studied in depth in the 1990s by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, among many others. In particular, they obtained the Plancherel formula for the L^2 space of G/H. An important aspect is that this generalizes the group case, obtained by Harish-Chandra, which corresponds to the case when G = G' x G' and H is the diagonal subgroup.

In our collaborative efforts with A. Afgoustidis, N. Higson, P. Hochs, Y. Song, we are studying this subject from the perspective of noncommutative geometry. I will describe this exciting new development, with a particular emphasis on describing what is new and how this is different from the traditional group case, i.e. the reduced group C*-algebra of G.

There has been considerable interest in the problem of whether every metric space of bounded geometry coarsely embeds into a uniformly convex Banach space due to the work of Kasparov and Yu that established a connection between such embeddings and the Novikov conjecture. Brown and Guentner were able to prove that a metric space with bounded geometry coarsely embeds into a reflexive Banach space. Kalton significantly extended this result to stable metric spaces and asked whether these classes are coarsely equivalent, i.e. whether every reflexive Banach space coarsely embeds into a stable metric space. Baudier introduced the notion of upper stability, a relaxation of stability, for metric spaces as a new invariant to distinguish reflexive spaces from stable metric spaces. In this talk, we show that in fact, every reflexive space is upper stable and also establish a connection of upper stability to the asymptotic structure of Banach spaces. This is joint work with F. Baudier and Th. Schlumprecht.

Named after mathematical physicists Kubo, Martin, and Schwinger, KMS states are a special class of states on any C$^*$-algebra admitting a continuous action of the real numbers. Unlike in the case of von Neumann algebras, where each modular flow has a unique KMS state, the collection of KMS states for a given flow on a C$^*$-algebra can be quite intricate. In this talk, I will explain what abstract properties these simplices have and show how one can realise such a simplex on various classes of simple C$^*$-algebras.

The study of topologically ordered quantum phases has led to interesting connections with, for example, the study of subfactors. In this talk, I will introduce a new axiomatisation of such quantum models defined on d-dimensional square lattices in terms of nets of projections. These local topological order axioms are satisfied by known 2D models such as the toric code and Levin-Wen models built on a unitary fusion category. We show that these axioms lead to a definition of boundary algebras naturally living on a hyperplane. This boundary algebra encodes information about the excitations in the bulk theory, leading to a bulk-boundary correspondence. I will outline the main points, with an emphasis on interesting connections to operator algebras and fusion categories. Based on joint work with C. Jones, Penneys, and Wallick (arXiv:2307.12552).

I will survey Cartan respectively diagonal subalgebras of nuclear C*-algebras. This setup corresponds to a presentation of the ambient C*-algebra as an amenable groupoid C*-algebra, which in turn means that there is an underlying structure akin to an amenable topological dynamical system.

The existence of such subalgebras is tightly connected to the UCT problem; the classification of Cartan pairs is largely uncharted territory. I will present new constructions of diagonals of the Jiang-Su algebra Z and of the Cuntz algebra O_2, and will then focus on distinguishing Cantor Cartan subalgebras of O_2.

A previously known function-theoretic characterisation of compactness for a weighted composition operator on BMOA is improved. Moreover, the same function-theoretic condition also characterises weak compactness and complete continuity. In order to close the circle of implications, the operator-theoretic property of fixing a copy of c0 comes in useful. 

I will discuss various definitions of quantum or noncommutative graphs that have appeared in the literature, along with motivating examples.  One definition is due to Weaver, where examples arise from quantum channels and the study of quantum zero-error communication.  This definition works for any von Neumann algebra, and is "spatial": an operator system satisfying a certain operator bimodule condition.  Another definition, first due to Musto, Reutter, and Verdon, involves a generalisation of the concept of an adjacency matrix, coming from the study of (simple, undirected) graphs.  Here we study finite-dimensional C*-algebras with a given faithful state; examples are perhaps less obvious.  I will discuss generalisations of the latter framework when the state is not tracial, and discuss various notions of a "morphism" of the resulting objects

A C*-algebra has the lifting property (LP) if any unital completely positive map into a quotient C*-algebra admits a completely positive lift. The local lifting property (LLP), introduced by Kirchberg in the early 1990s, is a weaker, local version of the LP.  I will present a method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C*-algebras of countable groups with (relative) property (T). This allows us to derive that the full C*-algebras of the groups $Z^2\rtimes SL_2(Z)$ and $SL_n(Z)$, for n>2, do not have the LLP. The same method allows us to prove that the full C*-algebras of a large class of groups with property (T), including those admitting a probability measure preserving action with non-vanishing second real-valued cohomology, do not have the LP.  In a different direction, we prove that the full C*-algebras of any non-finitely presented groups with property (T) do not have the LP. Time permitting, I will also discuss a connection with the notion of Hilbert-Schmidt stability for countable groups. This is based on a joint work with Pieter Spaas and Matthew Wiersma.

A self-similar group is a group $G$ acting on a regular, infinite rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups, like Grigorchuk's 2-group of intermediate growth, are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims, and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk, we discuss a result of the speaker and Benjamin Steinberg characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

Lie groupoids are closely connected to pseudo-differential calculus. On a vector bundle considered as a `commutative Lie groupoid' (i.e. as a family of commutative Lie groups), they can be treated using the Fourier transform. In this talk, we explore the extension of this idea to the noncommutative space by employing the tubular neighborhood construction and subsequently adopting a global approach through the introduction of deformation to the normal cone (groupoid). By utilizing this groupoid, we can construct the analytic index of pseudo-differential operators without relying on pseudo-differential calculus.

Furthermore, through the canonical construction of the space of functions with Schwartz decay, pseudo-differential operators on a manifold can be represented as an integral associated with smooth functions on the deformation to the normal cone. This perspective provides a geometric characterization that allows for the direct proof of fundamental properties of pseudo-differential operators.

In 1986, Katznelson and Tzafriri proved that, if $T$ is a power-bounded operator on a Banach space $X$, and the spectrum of $T$ meets the unit circle only at 1, then $\|T^n(I-T)\| \to 0$ as $n\to\infty$. Actually, they went further and proved that $\|T^nf(T)\| \to 0$ if $T$ and $f$ satisfy certain conditions. Soon afterward, analogous results were obtained for bounded $C_0$-semigroups $(T(t))_{t\ge0}$. Further extensions and variants were proved later. I will speak about several extensions to the Katznelson-Tzafriri theorem(s), including in particular a recent result(s) obtained by David Seifert and myself.

Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T). 

I will present several rigidity results for von Neumann algebras of wreath-like product groups. We show that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T). For a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an icc property (T) group G. These results are joint with Ionut Chifan, Denis Osin and Bin Sun.  

Time permitting, I will mention an additional application of wreath-like products obtained in joint work with Ionut Chifan and Daniel Drimbe, and showing that any separable II_1 factor is contained in one with property (T). This provides an operator algebraic counterpart of the group theoretic fact that every countable group is contained in one with property (T).

The pioneering work of Murray and von Neumann shows that any countable discrete group G gives rise in a canonical way to a group von Neumann algebra, denoted L(G). A main theme in operator algebras is to classify group von Neumann algebras, and hence, to understand how much information does L(G) remember of the underlying group G. In the amenable case, the classification problem is completed by the work of Connes from 1970s asserting that for all infinite conjugacy classes amenable groups, their von Neumann algebras are isomorphic.

In sharp contrast, in the non-amenable case, Popa's deformation rigidity/theory (2001) has led to the discovery of several instances when various properties of the group G are remembered by L(G). The goal of this talk is to survey some recent progress in this direction.

The notion of amenable actions by discrete groups on C*-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed until 2020. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also present far reaching results towards the weak containment problem which asks wether an action $\alpha:G\to \Aut(A)$ is amenable if and only if the maximal and reduced crossed products coincide.

In this lecture we report on joint work with Alcides Buss and Rufus Willett.

An anomalous symmetry of an operator algebra A is a mapping from a group G into the automorphism group of A which is multiplicative up to inner automorphisms. To any anomalous symmetry, there is an associated cohomology invariant in H^3(G,T). In the case that A is the Hyperfinite II_1 factor R and G is amenable, the associated cohomology invariant is shown to be a complete invariant for anomalous actions on R by the work of Connes, Jones, and Ocneanu.

In this talk, I will introduce anomalous actions from the basics discussing examples and the history of their study in the literature. I will then discuss two obstructions to possible cohomology invariants of anomalous actions on simple C*-algebras which arise from considering K-theoretic invariants of the algebras. One of the obstructions will be of algebraic flavour and the other will be of topological flavour. Finally, I will discuss the classification question for certain classes of anomalous actions.

abstract-Oxford.pdf

There has been a great deal of interest in understanding the link between the axiomatic descriptions of conformal field theory given by vertex operator algebras and conformal nets. In recent work, we establish an equivalence between certain vertex algebras and conformally-symmetric quantum field theories in the sense of Wightman. In this talk I will give an overview of these results and discuss some of the difficulties that arise, the functional analytic properties of vertex algebras, and some of the ideas for future work in this area.

This is joint work with James Tener and Yoh Tanimoto.

The Hilbert transform H is a basic example of a Fourier multiplier, and Riesz proved that H is a bounded operator on Lp(T) for all p between 1 and infinity.  We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative L_p spaces. The pioneering work in this direction is due to Mei and Ricard who proved L_p-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.  This is joint work with Adrián González and Javier Parcet.

By Jewett-Krieger theorems minimal dynamical systems on the Cantor set are topological analogous of ergodic systems on probability Lebesgue spaces. In this analogy and to study a Cantor minimal system, indicator functions of clopen sets (as continuous integer or real valued functions) are considered while they are mod out by the subgroup of all co-boundary functions. That is how dimension group which is an operator algebraic object appears in dynamical systems. In this talk, I try to explain a bit more about dimension groups from dynamical point of view and how it relates to topological factoring and spectrum of Cantor minimal systems.

The joint spectral radius of a finite family S of matrices measures the rate of exponential growth of the maximal norm of an element from the product set S^n as n grows. This notion was introduced by Rota and Strang in the 60s. It arises naturally in a number of contexts in pure and applied mathematics. I will discuss its basic properties and focus on a formula of Berger and Wang and results of J. Bochi that extend to several matrices the classical for formula of Gelfand that relates the growth rate of the powers of a single matrix to its spectral radius. I give new proofs and derive explicit estimates with polynomial dependence on the dimension, refining these results. If time permits I will also discuss connections with the Tits alternative, the notion of joint spectrum, and a geometric version of these results regarding groups acting on non-positively curved spaces.